Download A mathematical model for cancer chemotherapy treatment with a

Optimizing Cancer Treatments under Tumor-Immune System
Interactions
Urszula Ledzewicz,
Department of Mathematics and Statistics,
Southern Illinois University Edwardsville, USA
We are considering a model that describes the interactions between cancer
and immune system as an optimal control problem with the action of a single
cytotoxic agent as the control. In the uncontrolled system there exist both
regions of benign and of malignant cancer growth that are separated by the
stable manifold of a saddle point. The aim of treatment is to move an initial
condition that lies in the malignant region into the region of benign growth.
A Bolza formulation of the objective is given that includes a penalty term
that approximates this separatrix by its tangent space and minimization of
the objective is tantamount to moving the state of the system across this
boundary.
In this talk, generalizing earlier results when the action of the cytotoxic
agent on the immune system was assumed negligible, we now consider a
killing action of the cytotoxic drug on both the cancer cells and the cells of
the immune system. For various values of a parameter ε that describes the
relative effectiveness of the drug onto these two classes of cells, we discuss
the structure of optimal and near-optimal controls that move the system into
the region of attraction of the benign stable equilibrium. In particular, the
existence and optimality of a singular arc will be addressed using Lie
bracket computations and the Legendre-Clebsch condition.